The focus of this research is to describe children's ways of quantifying length and perimeter in response to an open-ended reasoning task. Twenty-five children from grade 2 through grade 10 were given two fixed perimeter tasks in a structured interview setting. A 1997 framework proposed by Clements, Battista, Sarama, Swaminathan, and McMillen was used for the study. Children's solution strategies suggested a modified framework consisting of levels 1a, lb, 2a, 2b, 3a and 3b, where the distinction of 3a and 3b emerged as a contrast between dynamic and static ways of coordinating and sequencing cases of measured shapes. Although more advanced strategies appeared among older children, some of the younger children nonetheless exhibited level 3 strategies and conversely, some grade 10 students seemed limited to level 2b strategies. The analysis also addresses the students' ways of coordinating representations of space and number, and ways of evaluating boundary cases. This study supports and elaborates upon previous studies based on a classroom curriculum implementation, and on teaching experiments, providing a broad account of developmental levels of knowledge for path length. The development of children's spatial reasoning has recently gained emphasis within the mathematics curriculum, for the U.S. Measurement system and geometry have been identified as two of five content-focused standards for K-12 instruction (NCTM, 2000). Furthermore, reasoning and proof retains a prominent position as one of the five process-focused standards, emphasizing children's ways of explaining and describing their responses to substantial tasks across the curriculum. Measurement and geometry tasks provide critical sites for helping children engage in reasoning toward proof. As the curriculum is shifted to emphasize measurement, especially in the first two of the four K-12 grade bands, there is a concurrent need to describe the increasingly sophisticated ways of measuring how children are expected to develop. If teachers are helped in connecting their own knowledge of children's thinking and strategies for measurement to instructional decisions, the expectation that teachers will provide appropriate sequences of instructional activities, ask relevant questions, and promote mathematically rich discussions. (Contains 44 references and 11 figures.) (Author/ASK)